This PCS is then used to form a waveguide (cf. [1]}) as shown in Fig. REF , except the interface hole radii are chosen to be smaller than \(r_2\)  to increase the initial distance between the edges of the interface holes. For our purposes, using the GME technique, the waveguide is modeled with a super cell \(a\)  wide in the \(x\)  direction. As the computation time increases with the size of the super cell, we chose a length of approximately \(13a\)  in the \(y\)  direction for the optimization calculations. To obtain more accurate calculations of the relevant PCS bands which define the PBG, such as those performed for Figs. REF  and REF , a super cell approximately \(30a\)  long in the \(y\)  direction is used. The super cell with width \(a\)  contains two holes which form the interface. The two interface holes' radii and position are allowed to be modified as well as the slab thickness, while enforcing the same geometric constraints as described above for the PC. The FOM used is the bandwidth-mid bandwidth ratio, which we define as the bandwidth divided by its central frequency value. We then used the stochastic optimization algorithm termed “Adam” [2]} with an objective function returning this FOM.
